What does < a^{\dagger} a > mean exactly? [Archive]

Yi-Zen Chu; Yiren Qu

Dec22-04, 05:58 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hello everyone\n\nI\'m trying to understand the use of density matrices in quantum field\ntheory - can anyone suggest a good reference (either online or offline)?\n\nWhat does < a^{\\dagger} a > mean exactly, in particular? I saw this in\nthe context of calculating the density matrix of neutrinos of a\nparticular momentum. One of the relations given was\n\n< a^{\\dagger}_j[p] a_i[p\'] > = (2 Pi)^{3} delta^{3}[p-p\'] (\\rho_{p})_ij\n\nwhere i,j are labels of the neutrino flavors, I believe.\n\nWhat can the bra and ket possibly mean in such a case? Momentum\neigenstates? What about flavor?\n\nThanks,\nYi-Zen\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hello everyone

I'm trying to understand the use of density matrices in quantum field
theory - can anyone suggest a good reference (either online or offline)?

What does < a^{\dagger} a > mean exactly, in particular? I saw this in
the context of calculating the density matrix of neutrinos of a
particular momentum. One of the relations given was

< a^{\dagger}_j[p] a_i[p'] > = (2 \Pi)^{3} \delta^{3}[p-p'] (\rho_{p})_ij

where i,j are labels of the neutrino flavors, I believe.

What can the bra and ket possibly mean in such a case? Momentum
eigenstates? What about flavor?

Thanks,
Yi-Zen

Arnold Neumaier

Dec23-04, 05:45 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Yi-Zen Chu; Yiren Qu wrote:\n> Hello everyone\n>\n> I\'m trying to understand the use of density matrices in quantum field\n> theory - can anyone suggest a good reference (either online or offline)?\n>\n> What does < a^{\\dagger} a > mean exactly, in particular?\n>\n> What can the bra and ket possibly mean in such a case?\n\nThere are no bras and kets in this formula.\na^{\\dagger} a is the number operator written in second-quantized form,\nand <.> is the vacuum expectation. Similar for your other example,\nexcept that the operator is more complicated.\n\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Yi-Zen Chu; Yiren Qu wrote:
> Hello everyone
>
> I'm trying to understand the use of density matrices in quantum field
> theory - can anyone suggest a good reference (either online or offline)?
>
> What does < a^{\dagger} a > mean exactly, in particular?
>
> What can the bra and ket possibly mean in such a case?

There are no bras and kets in this formula.
a^{\dagger} a is the number operator written in second-quantized form,
and <.> is the vacuum expectation. Similar for your other example,
except that the operator is more complicated.

Arnold Neumaier

meyr2@web.de

Dec23-04, 05:46 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hi,\n\nwhen looking at the second formula, that looks like the left side of a\ncommutator between a^dagger and a, it looks like that\n\n<a^+ a> is just the expectation value and that someone just calculated\n\n< a^+ a> = Tr (rho a^+ a)\nAt least a^+ a is the number operator for some mode.\n\nRene.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hi,

when looking at the second formula, that looks like the left side of a
commutator between a^{dagger} and a, it looks like that

<a^+ a> is just the expectation value and that someone just calculated

< a^+ a> = Tr (\rho a^+ a)
At least a^+ a is the number operator for some mode.

Rene.

Yi-Zen Chu; Yiren Qu

Dec25-04, 03:25 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Arnold Neumaier wrote:\n>\n> There are no bras and kets in this formula.\n> a^{\\dagger} a is the number operator written in second-quantized form,\n> and <.> is the vacuum expectation. Similar for your other example,\n> except that the operator is more complicated.\n\nHello Arnold\n\nWhen you say vacuum expectation, don\'t you mean <0| a^{\\dagger}_j[p]\na_i[p\'] |0>, where |0> means 0 particles at a given momentum? If so\nwouldn\'t this just be identically zero?\n\nI may not be understanding what exactly you mean by no bras and kets.\n\nIs it possible to take expectation values with respect to non-ground\nstates? Say with respect to the 1 muon neutrino state at a given momentum?\n\nThanks!\n\nYi-Zen\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Arnold Neumaier wrote:
>
> There are no bras and kets in this formula.
> a^{\dagger} a is the number operator written in second-quantized form,
> and <.> is the vacuum expectation. Similar for your other example,
> except that the operator is more complicated.

Hello Arnold

When you say vacuum expectation, don't you mean <0| a^{\dagger}_j[p]a_i[p'] |0>, where |0> means particles at a given momentum? If so
wouldn't this just be identically zero?

I may not be understanding what exactly you mean by no bras and kets.

Is it possible to take expectation values with respect to non-ground
states? Say with respect to the 1 muon neutrino state at a given momentum?

Thanks!

Yi-Zen

Yi-Zen Chu; Yiren Qu

Dec25-04, 03:30 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>fruit salad and sparkling water.\n\n\n\nBébé Buffet 1\n\nShow off with whole roasted children replete with apples in mouths -\nand babies? heads stuffed with wild rice. Or keep it simple with a\nhearty main course such as stew, lasagna, or meat loaf.\n\nSome suggestions\n\nPre-mie pot pies, beef stew, leg of lamb, stuffed chicken, roast pork spiral ham,\nCranberry pineapple salad, sweet potatoes in butter, vegetable platter, tossed salad with tomato and avocado, parsley new potatoes, spinich cucumber salad, fruit salad\nBran muffins, dinner rolls, soft breadsticks, rice pilaf, croissants\nApple cake with rum sauce, frosted banana nut bread sherbet, home made brownies\nIced tea, water, beer, bloody marys, lemonade, coffee\n\nThe guests select food, beverages, silverware... everything from the buffet table.\nThey move to wherever they are comfortable, and sit with whoever they choose.\nProvide trays so your guests will not spill everything all over your house from\ncarrying too much, nor will they have to make 10 trips back and fourth from the\nservice stations.\n\n\n\nRoast Leg of Amputee\n\nBy all means, substitute lamb or a good beef roast if the haunch\nit is in any way diseased. But sometimes surgeons make mistakes,\nand if a healthy young limb is at hand, then don?t hesitate to cook\nit to perfection!\n\n1 high quality limb, rack, or roast\nPotatoes, carrot\nOil\ncelery\nonions\ngreen onions\nparsley\ngarlic\nsalt, pepper, etc\n2 cups beef stock\n\nMarinate meat (optional, not necessary with better cuts).\nSeason liberally and lace with garlic cloves by making incisions,\nand placing whole cloves deep into the meat.\nGrease a baking pan, and fill with a thick bed of onions,\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>fruit salad and sparkling water.

Bébé Buffet 1

Show off with whole roasted children replete with apples in mouths -
and babies? heads stuffed with wild rice. Or keep it simple with a
hearty main course such as stew, lasagna, or meat loaf.

Some suggestions

Pre-mie pot pies, beef stew, leg of lamb, stuffed chicken, roast pork spiral ham,
Cranberry pineapple salad, sweet potatoes in butter, vegetable platter, tossed salad with tomato and avocado, parsley new potatoes, spinich cucumber salad, fruit salad
Bran muffins, dinner rolls, soft breadsticks, rice pilaf, croissants
Apple cake with rum sauce, frosted banana nut bread sherbet, home made brownies
Iced tea, water, beer, bloody marys, lemonade, coffee

The guests select food, beverages, silverware... everything from the buffet table.
They move to wherever they are comfortable, and sit with whoever they choose.
Provide trays so your guests will not spill everything all over your house from
carrying too much, nor will they have to make 10 trips back and fourth from the
service stations.

Roast Leg of Amputee

By all means, substitute lamb or a good beef roast if the haunch
it is in any way diseased. But sometimes surgeons make mistakes,
and if a healthy young limb is at hand, then don?t hesitate to cook
it to perfection!

1 high quality limb, rack, or roast
Potatoes, carrot
Oil
celery
onions
green onions
parsley
garlic
salt, pepper, etc
2 cups beef stock

Marinate meat (optional, not necessary with better cuts).
Season liberally and lace with garlic cloves by making incisions,
and placing whole cloves deep into the meat.
Grease a baking pan, and fill with a thick bed of onions,

Yi-Zen Chu; Yiren Qu

Dec25-04, 03:34 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>will need one for the top also).\nBoil the prepared delicacy until the meat starts to come off the bones.\nRemove, de-bone and cube; continue to reduce the broth.\nBrown the onions, peppers and celery.\nAdd the meat then season, continue browning.\nDe-glaze with sherry, add the reduced broth.\nFinally, put in the root vegetables and simmer for 15 minutes.\nAllow to cool slightly.\nPlace the pie pan in 375 degree oven for a few minutes so bottom crust is not soggy,\nreduce oven to 325.\nFill the pie with stew, place top crust and with a fork, seal the crusts together\nthen poke holes in top.\nReturn to oven and bake for 30 minutes, or until pie crust is golden brown.\n\n\n\nSudden Infant Death Soup\n\nSIDS: delicious in winter, comparable to old fashioned Beef and Vegetable Soup.\nIts free, you can sell the crib, baby clothes, toys, stroller... and so easy to\nprocure if such a lucky find is at hand (just pick him up from the crib and\nhe?s good to go)!\n\nSIDS victim, cleaned\n½ cup cooking oil\nCarrots\nonions\nbroccoli\nwhole cabbage\nfresh green beans\npotato\nturnip\ncelery\ntomato\n½ stick butter\n1 cup cooked pasta (macaroni, shells, etc.)\n\nRemove as much meat as possible, cube, and brown in hot oil.\nAdd a little water, season, then add the carcass.\nSimmer for half an hour keeping the stock thick.\nRemove the carcass and add the vegetables slowly to the stock,\nso that it remains boiling the whole time.\nCover the pot and simmer till vegetables are tender\n(2 hours approximately).\nContinue seasoning to taste.\nBefore serving, add butter and pasta,\nserve piping with hot bread and butter.\n\n\n\nOffspring Rolls\n\nSimilar to Vietnamese style fried rolls, they have lots of meat\n(of course this can consist of chicken, beef, pork, or shrimp).\nWho can resist this classic appetizer; or light lunch served with\na fresh salad? Versatility is probably this recipe?s greatest virtue\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>will need one for the top also).
Boil the prepared delicacy until the meat starts to come off the bones.
Remove, de-bone and cube; continue to reduce the broth.
Brown the onions, peppers and celery.
Add the meat then season, continue browning.
De-glaze with sherry, add the reduced broth.
Finally, put in the root vegetables and simmer for 15 minutes.
Allow to cool slightly.
Place the pie pan in 375 degree oven for a few minutes so bottom crust is not soggy,
reduce oven to 325.
Fill the pie with stew, place top crust and with a fork, seal the crusts together
then poke holes in top.
Return to oven and bake for 30 minutes, or until pie crust is golden brown.

Sudden Infant Death Soup

SIDS: delicious in winter, comparable to old fashioned Beef and Vegetable Soup.
Its free, you can sell the crib, baby clothes, toys, stroller... and so easy to
procure if such a lucky find is at hand (just pick him up from the crib and
he?s good to go)!

SIDS victim, cleaned
½ cup cooking oil
Carrots
onions
broccoli
whole cabbage
fresh green beans
potato
turnip
celery
tomato
½ stick butter
1 cup cooked pasta (macaroni, shells, etc.)

Remove as much meat as possible, cube, and brown in hot oil.
Add a little water, season, then add the carcass.
Simmer for half an hour keeping the stock thick.
Remove the carcass and add the vegetables slowly to the stock,
so that it remains boiling the whole time.
Cover the pot and simmer till vegetables are tender
(2 hours approximately).
Continue seasoning to taste.
Before serving, add butter and pasta,
serve piping with hot bread and butter.

Offspring Rolls

Similar to Vietnamese style fried rolls, they have lots of meat
(of course this can consist of chicken, beef, pork, or shrimp).
Who can resist this classic appetizer; or light lunch served with
a fresh salad? Versatility is probably this recipe?s greatest virtue

Arnold Neumaier

Dec27-04, 07:32 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Yi-Zen Chu; Yiren Qu wrote:\n> Arnold Neumaier wrote:\n>\n>>There are no bras and kets in this formula.\n>>a^{\\dagger} a is the number operator written in second-quantized form,\n>>and <.> is the vacuum expectation. Similar for your other example,\n>>except that the operator is more complicated.\n>\n> When you say vacuum expectation, don\'t you mean <0| a^{\\dagger}_j[p]\n> a_i[p\'] |0>, where |0> means 0 particles at a given momentum? If so\n> wouldn\'t this just be identically zero?\n\nSorry for my nonsense. I wasn\'t paying close enough attention.\n<.> is the expectation of an _arbitrary_ state. Of course, in the\nvacuuum state, the number of particles is zero, as you just observed.\n\n\n> Is it possible to take expectation values with respect to non-ground\n> states? Say with respect to the 1 muon neutrino state at a given momentum?\n\nYes, of course. And also to mixed states.\n\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Yi-Zen Chu; Yiren Qu wrote:
> Arnold Neumaier wrote:
>
>>There are no bras and kets in this formula.
>>a^{\dagger} a is the number operator written in second-quantized form,
>>and <.> is the vacuum expectation. Similar for your other example,
>>except that the operator is more complicated.
>
> When you say vacuum expectation, don't you mean <0| a^{\dagger}_j[p]
> a_i[p'] |0>, where |0> means particles at a given momentum? If so
> wouldn't this just be identically zero?

Sorry for my nonsense. I wasn't paying close enough attention.
<.> is the expectation of an _arbitrary_ state. Of course, in the
vacuuum state, the number of particles is zero, as you just observed.

> Is it possible to take expectation values with respect to non-ground
> states? Say with respect to the 1 muon neutrino state at a given momentum?

Yes, of course. And also to mixed states.

Arnold Neumaier

mikem@despammed.com

Jan9-05, 07:46 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nYi-Zen Chu; Yiren Qu wrote:\n\n>> Is it possible to take expectation values with\n>> respect to non-ground states? Say with respect\n>> to the 1 muon neutrino state at a given momentum?\n\nArnold Neumaier wrote:\n\n> Yes, of course. [...]\n\nPerhaps I\'m being overly pedantic, but I\nthink one must take care for the particular\ncase mentioned. "Muon neutrino" suggests\na flavour eigenstate of the weak interaction,\nbut "definite momentum" suggests a\nmass eigenstate. Blasone, Vitiello et al have\nstudied [ref 1] the QFT of neutrino oscillations\nand concluded that the flavour eigenstates\nand mass eigenstates live in different unitarily\ninequivalent representations. Hence, the\nparticular state mentioned above does not\nexist in the familiar sense.\n\n[Ref 1]: hep-th/9501263.\nSee also the review thesis of\nCapolupo hep-th/0408228.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Yi-Zen Chu; Yiren Qu wrote:

>> Is it possible to take expectation values with
>> respect to non-ground states? Say with respect
>> to the 1 muon neutrino state at a given momentum?

Arnold Neumaier wrote:

> Yes, of course. [...]

Perhaps I'm being overly pedantic, but I
think one must take care for the particular
case mentioned. "Muon neutrino" suggests
a flavour eigenstate of the weak interaction,
but "definite momentum" suggests a
mass eigenstate. Blasone, Vitiello et al have
studied [ref 1] the QFT of neutrino oscillations
and concluded that the flavour eigenstates
and mass eigenstates live in different unitarily
inequivalent representations. Hence, the
particular state mentioned above does not
exist in the familiar sense.

[Ref 1]: http://www.arxiv.org/abs/hep-th/9501263.
See also the review thesis of
Capolupo http://www.arxiv.org/abs/hep-th/0408228.